Wencheng Huang

To clarify the mechanism of risk evolution in railway transportation system (RTS), swallowtail catastrophe is applied to analyze the risk state changes. Firstly, the Duffing equation is used to characterize the system. Based on historical accident data fitting results, we describe the dynamic changes of system damping and external excitation force over time when risk factors in the system, deduct the equilibrium surface function and singularity set, and establish the nonlinear state change equation of swallowtail catastrophe. Then, the mathematical characteristics of three-dimensional bifurcation set is analyzed. Based on mathematical analysis, we can find that: 1)balanced points appear or disappear in pairs; 2) bifurcation set can be classified into five parts according to the number of minimum, maximum and singular points; and 3) catastrophe of the system state can only occur on the space surface. Compared with the cusp catastrophe model commonly used in the transportation field, our paper constructed a three-dimensional control space containing three risk factors, thereby enabling a more accurate description of the system complexity under the coupling effect of multiple risk factors. Next, the energy measured by risk pulse theory is introduced to formulate risk factors base on real-world engineering experiment data, and system state change mechanism is analyzed. A case study is conducted by taking the RTS in China as the background, the results show that: when the system state control point runs from the area with more stable singularities to that with less stable singularities, and its trajectory crosses the bifurcation set, the properties of the stable equilibrium point will change, the system state will also change. When applying swallowtail catastrophe, the state changes of a RTS satisfy bimodality, inaccessibility, sudden transitions and divergence, but don’t satisfy hysteresis.

This paper examines the structural and reliability aspects of multilayer transportation networks. Using Space L, we construct these networks and develop a topology index system including node overlap degree, node activity degree, average path length, network diameter, clustering coefficient, betweenness, betweenness centrality, multilayer participation coefficient, edge overlap coefficient, and edge crossing coefficient. We evaluate reliability by analyzing node and edge failures. A case study demonstrates our method effectively captures node/edge heterogeneity and topological features in integrated transportation systems. Results show the metro-urban bus multilayer network exhibits small-world and scale-free properties. Nodes with high overlap/activity degree, betweenness, or participation coefficient significantly impact the network. Paths with larger edge overlap coefficients offer redundancy, faster recovery, and higher reliability post-failure. Networks with greater edge crossing coefficients exhibit enhanced redundancy and risk resistance.